Answer:
a. y is the response variable. There are 3 explanatory variables, namely x1, x2 and x3.
b. y = 1.6 + 3.5 x 2 - 7.9 x 1 + 2 x 5 = 10.7
c. 90% confidence interval for the coefficient of x1 = (2.721, 4.279)
d. The claim is true. x1 has predictability towards y.
Step-by-step explanation:
c. Use the following four-step approach to construct a confidence interval.
- Identify a sample statistic. From the regression equation, we see that the coefficient of x1 is 3.5
- Select a confidence level. We are working with a 90% confidence level.
- Find the margin of error
• Find standard error for the coefficient of x1. The standard error is given as 0.419
• Find critical value. The critical value is a t score with degrees of freedom equal to n - k (n= number of data points, k = number of parameters). To find the critical value, we take these steps.
o Compute alpha (α):
α = 1 - (confidence level / 100)
α = 1 - 90/100 = 0.1
o Find the critical probability (p*):
p* = 1 - α/2 = 1 - 0.1/2 = 0.95
o Find the degrees of freedom (df):
df = n - k = 12 - 4 = 8.
o The critical value is the t statistic having 8 degrees of freedom and a cumulative probability equal to 0.95. From the t Distribution Calculator, we find that the critical value is 1.86.
• Compute margin of error (ME):
ME = critical value * standard error
ME = 1.86 * 0.419 = 0.779
Specify the confidence interval. The range of the confidence interval is defined by the sample statistic + margin of error. And the uncertainty is denoted by the confidence level.
Therefore, the 90% confidence interval for the coefficient of x1 is 3.5 ± 0.779, which is 2.721 to 4.279
.
d. 5% level of significance is conversely translates to a 95% level of confidence. Using the same approach, we identify the 95% confidence interval for the coefficient of x1 is 3.5 ± 0.966, which is 2.534 to 4.466
.
0 falls out of the confidence interval, therefore the claim that the coefficient of x1 is different from 0. It means x1 has predictability towards y.
